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Preconditioned conjugate gradients for solving singular systems. (English) Zbl 0659.65031
Sparse singular systems have to be solved for instance with semi-definite Neumann-problems. Such systems \(Ax=b\) with a rank deficiency of one can be solved by fixing a \(x_ i\), deleting the corresponding row and column, adjusting the right hand side and solving the new system with a preconditioned conjugate gradient method. Alternatively one may solve \(Ax=b_ R\) with \(b_ R=b-{\mathcal P}_{N(A)}b\) with the same method, if the kernel N(A) is known explicitly. The author shows, that this method often is faster than the first one. Conditions for the existence of a nonsingular incomplete Cholesky decomposition are given. The results are illustrated by a numerical experiment.
Reviewer: N.Köckler

MSC:
65F10 Iterative numerical methods for linear systems
65F05 Direct numerical methods for linear systems and matrix inversion
65F50 Computational methods for sparse matrices
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
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